Existence and Stability Results of a Nonlinear Fractional Integro-Differential Equation with Integral Boundary Conditions

Download PDF


DOI: 10.46793/KgJMat2205.685A


This paper deals with the stability results for solution of a fractional integro-differential problem with integral conditions. Using the Krasnoselskii’s, Banach fixed point theorems, we proof the existence and uniqueness results. Based on the results obtained, conditions are provided that ensure the generalized Ulam stability of the original system. The results are illustrated by an example.


Fractional integro-differential equation, existence, stability, nonlocal conditions, fixed point theory, single valued maps.


[1]   A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud. 204, Elsevier, Amsterdam, 2006.

[2]   B. Ahmad, Y. Alruwaily, A. Alsaedi and S. K. Ntouyas, Existence and stability results for a fractional order differential equation with non-conjugate Riemann-Stieltjes integro-multipoint boundary conditions, Mathematics 7(249) (2019), 1–14.

[3]   B. Ahmad, A. Alsaedi, S. Salem and S. K. Ntouyas, Fractional differential equation involving mixed nonlinearities with nonlocal multi-point and Reimann-Steiljes integral-multi-strip conditions, Fractal and Fractional 3(34) (2019), 1–16.

[4]   S. B. Hadid, Local and global existence theorems on differential equations of non-integer order, J. Fract. Calculus 7 (1995), 101–105.

[5]   S. M. Momani, Local and global uniqueness theorems on differential equations of non-integer order via Bihari’s and Gronwall’s inequalities, Revista Técnica de la Facultad de Ingeniería Universidad del Zulia 23(1) (2000), 66–69.

[6]   I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, Londin, Toronto, 1999.

[7]   S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993.

[8]   S. M. Momani, S. Hadid and Z. M. Alawenh, Some analytical properties of solutions of differential equations of non-integer order, Int. J. Math. Math. Sci. 13 (2004), 697–701.

[9]   A. J. S. Momani and S. Al-Azawi, Local and global uniqueness theorems on fractional integro-differential equations via Bihari’s and Gronwall’s inequalities, Soochow Journal of Mathematics 33(4) (2007), 619–627.

[10]   V. V. Anh and R. Mcvinish, Fractional differential equations driven by Levy noise, Journal of Applied Mathematics and Stochastic Analysis 16(2) (2003), 97–119.

[11]   Y. Zhou, Existence results for fractional neutral integro-differential equations with state-dependent delay, Comput. Math. Appl. 62 (2011), 1275–1283.

[12]   Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014.